Deformations of Algebraic Schemes

Edoardo Sernesi - vita Present position: Professore ordinario di Geometria, FacoltA di Scienze MFN, UniversitA Roma Tre Education: - Laurea in Matematica- UniversitA di Roma, 1969 - Ph.D. in Mathematics - Brandeis University, 1976 Professional experience: - Assistente ordinario di Geometria, UniversitA di Ferrara, 1974-1980. - Professore straordinario di Geometria UniversitA di Roma "La Sapienza", 1980-1983. - Professore ordinario di Geometria UniversitA di Roma "La Sapienza", 1983-1992. - Professore ordinario di Geometria UniversitA Roma Tre, from 1992.

• Introduction.- Infinitesimal Deformations: Extensions. Locally Trivial Deformations.- Formal Deformation Theory: Obstructions. Extensions of Schemes. Functors of Artin Rings. The Theorem of Schlessinger. The Local Moduli Functors.- Formal Versus Algebraic Deformations. Automorphisms and Prorepresentability.- Examples of Deformation Functors: Affine Schemes. Closed Subschemes. Invertible Sheaves. Morphisms.- Hilbert and Quot Schemes: Castelnuovo-Mumford Regularity. Flatness in the Projective Case. Hilbert Schemes. Quot Schemes. Flag Hilbert Schemes. Examples and Applications. Plane Curves.- Appendices: Flatness. Differentials. Smoothness. Complete Intersections. Functorial Language.- List of Symbols.- Bibliography.

From the reviews: "One of the goals of Springer's Grundlehren series is to provide reliable and thorough accounts of certain portions of mathematics. This volume by Edoardo Sernesi does just that, and hence fits the series well. ... So this is a book for algebraic geometers; for them, it'll prove to be a useful resource and reference." (Fernando Q. Gouvea, MathDL, August, 2006) "Without any doubt, this is a masterly book on a highly advanced topic in algebraic geometry. ... The entire text is kept at a level that makes it suitable for graduate students ... . But even for experts and active researchers in algebraic geometry, this unique book on algebraic deformation theory offers a great deal of inspiration and new insights, too, and its future role as a standard source and reference book in the field can surely be taken for granted from now on." (Werner Kleinert, Zentralblatt MATH, Vol. 1102 (4), 2007) "The book under review gives an introduction to classical deformation theory using modern language, and is apparently unique among textbooks in the recent literature in that it is largely self-contained and covers the main topics ... . It will be attractive for graduate students with a basic knowledge of commutative algebra and algebraic geometry as a base for advanced lectures. The need for such a book was evident for a long time; the reviewer is happy to have it on his bookshelf." (Marko Roczen, Mathematical Reviews, Issue 2008 e)